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A Multi-objective Approach for the Combined Master
Surgical Schedule and Surgical Case Assignment
Problems
Salma Makboul, Said Kharraja, Abderrahman Abbassi, Ahmed El Hilali
Alaoui
To cite this version:
A Multi-objective Approach for the Combined Master Surgical Schedule
and Surgical Case Assignment Problems
Salma MAKBOUL MCS Laboratory FSTF USMBA Fez, MOROCCO salma.makboul@usmba.a c.ma Said KHARRAJA LASPI Univ-Lyon, Univ Saint
Etienne Roanne, FRANCE said.kharraja@univ-st-etienne.fr Abderrahman ABBASSI Faculty of Sciences Semlalia Cadi Ayyad University Marrakech, MOROCCO abbas-si.abdarrahman@gmail.c om Ahmed EL HILALI ALAOUI Euromed University Fez, MOROCCO a.elhilali-alaoui@ueuromed.org
ABSTRACT: In this study, we propose a multi-objective approach to address the Master Surgical Schedule (MSS) and
the Surgical Case Assignment (SCAP) problems for the common Operating Theater (OT) in an integrated hospital facil-ity. The approach accounts for both surgeons’ and Operating Rooms’ (OR) availability and restrictions. We propose in this paper a multi-objective programming model that supports OT decision making to ensure patients' and surgeons’ satisfaction and hospital quality of service by respecting surgeries due dates and balancing surgeons’ workloads. The proposed approach determines the surgical discipline to perform on each session, the surgical cases assigned on each session, and the operations’ start time on a weekly basis. The -constraint method is used to solve the multi-objective problem. The computational experiments are performed using ILOG CPLEX optimization studio. The model is tested on the empirical data archives of a medium-size French hospital and show the quality of solutions achieved using the pro-posed approach.
KEYWORDS
:
Operating Theater Planning, Multi-objective Programming, Master Surgical Schedule, Surgical CaseAssignment, epsilon-constraint method.
1 INTRODUCTION
The OT is both the cost and revenue center of the hospi-tal (Zhu et al., 2018). Efficiently managing its resource can strongly impact the quality of health service.
Given patients waiting list, OR characteristics and sur-geons’ timetable, the planning consists of deciding the scheduling of patients to be performing during the plan-ning horizon according to multiple performances factors such as ORs’ utilization, patients’ due dates, throughput, etc.
The surgeries are performed within the OR sessions, an OR session is a half or full day. The OT problem are studied under three strategies, the open scheduling strat-egy, the block scheduling strategy and the modified block scheduling strategy.
The block scheduling strategy is to allocate the OR to different surgeons or group and it is more performed in European hospital. Its advantages are reducing planning complexity. However, the block scheduling strategy may present some issues, since the ORs session can only be allocated to one surgeon, even if the surgeon does not perform any surgical case. This is the reason why the modified block scheduling is introduced (Younespour et
al., 2019). The open scheduling strategy is more flexible than the block scheduling strategy because no pre-specified session to discipline assignment exists (Agnetis
et al., 2014). In the open scheduling strategy the schedul-ing follows the principle of first-come first-served (Au-gusto et al., 2010). It is more performed in the American hospitals (Zhu et al., 2018).
The OT managers face complex problem to give an efficient planning, the problems include:
Deciding which surgical specialty to perform on each OR session
Assigning the patients to the OR session ac-cording to several factors
Sequencing the elective patients on each OR session
The first problem is tactical; it involves the Master Sur-gical Schedule problem (MSS). The second and third problems are operational; the second is the Surgical Case Assignment problem (SCAP) and the third problem is the Elective Surgery Sequencing problem (ESS).
In this paper, we present a new approach to support OT management; we define the MSS on a weekly basis and allocate surgeries to ORs with taking into account re-sources’ availability and various OT restrictions. Balanc-ing surgeons’ workload is one of the important factors to ensure surgeons’ satisfaction and provide a better quality of service.
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Given the multi-objective nature of the problem, the -constraints method is applied to reach solutions from the Pareto Front using different values.
The reminder of this paper is as follows. The literature review of OT management related works is presented in Section 2. The problem is described in detail in Section 3. Mathematical formulation is presented and described in Section 4. Section 5 presents the computational results of the proposed approach. Finally, Conclusions and perspectives are given in Section 6.
2 LITERATURE REVIEW
In past years, OT management and scheduling was re-viewed by several researchers.
The decision levels are organized into three levels: long-term strategic, medium-long-term tactical and short-long-term operational.
At the strategic level, the problem involves capacity planning, capacity allocation, and case-mix problems. The decisions made at the strategic level are applied in a planning horizon of several months (Blake et al. 2002). (Ma and Demeulemeester, 2013) proposed an approach to combine capacity allocation with case-mix. (Creemers et al., 2012) studied the problem of allocating service time slots to different patient classes to minimize the total expected waiting time of a patient.
The tactical level involves the Master Surgical Schedul-ing problem (MSS) and it is the problem under study in this paper. The MSS is a cyclic timetable; it allocates the surgical disciplines to OR under their requirements. (Cardoen et al., 2009) defined a MSS that decides the amount and type of available ORs, the open hours, and the surgeons. (Agnetis et al., 2014) introduced a decom-position approach to address the MSS and SCAP by assigning the disciplines to the available sessions first and then allocate surgeries to them on a weekly basis. (Adan et al., 2009) proposed an approach to generate a MSS that aims to optimize the utilization of resources by considering stochastic length of stay.
The operational level involves the Surgery Scheduling Problem (SSP) following tactical level decisions. The SSP problem is divided into two steps; the first step is the advance scheduling, referred in the literature as the Surgical Case Assignments problem (SCAP) and alloca-tion scheduling problem. The SCAP is to assign an OR and a day to surgeries, while the allocation schedule defines the start time of surgeries.
(Dios et al. 2015) introduced a decision support tool to deal with the surgery advance scheduling problem. (Vancroonenburg et al., 2015) proposed an approach to solve the allocation scheduling problem following the advance scheduling step.
Some authors deal with both advanced and allocation problems (Aringhieri et al., 2015, Marques et al., 2012, Riise and Burke, 2011).
Several solution approaches have been proposed to deal with multi-objective problems. Exact methods compute the entire Pareto Front while the heuristic search for
solutions close to Pareto-optimal solutions (Abounacer et al., 2014) (Ehrgott and Gandibleux, 2002).
One of the classical methods to handle a multi-objective problem is the -constraint method. This approach was introduced by (Haimes et al., 1971) and defined as one of the objective functions that are selected to be opti-mized while the other(s) are added as an additional con-straint in the optimization problem.
The proposed approach in this study is a bi-objective problem solved using the -constraint method and aims to satisfy the surgeons and patients and ensure a good quality of service. It is considered as a combination be-tween the MSS and the SSP that handles both SCAP and allocation problem. The proposed approach accounts for the availability of both human and material resources.
3 PROBLEM DESCRIPTION
This study addresses OT management, we decide which specialty to be performed on ORs session and allocate surgical cases to them with taking into account OT re-strictions. Each surgical case has:
Processing time (duration of the surgical cases, we assume that the surgical cases are determin-istic)
Due date (the nominal date for the surgical case to be performed)
Surgery discipline to be assigned to.
The MSS is subject to various restrictions that must be taking into consideration:
Discipline-to-OR restrictions. Some surgical disciplines can only be performed in a restricted set of ORs.
Limits on discipline parallelism. A limited number of ORs session of the same discipline can take place at the same time
OR sessions-per-discipline restrictions. Lower and upper limits to the number of OR sessions assigned
OR reservation. The hospital management may reserve some OR sessions to certain surgical disciplines
We assume that surgeons have a timetable and are not available all days in the planning horizon.
In the following model, we assign a dummy patient (1) to each OR session. Each session starts and ends with the dummy patient 1 to simplify modeling and sequencing.
4 MATHEMATICAL MODELING
4.1 Parameters
: Set of surgical disciplines
: Waiting list of surgeries in discipline
: Set of surgeons in discipline
: Days to the due-date (nominal date) for patient of discipline
: Expected duration of patient of discipline : Set of the operating rooms
: The score for surgery of discipline : OR session capacity
: Set of days
: Maximum number of OR session for discipline
: Minimum number of OR session for discipline
: Maximum number of parallel OR session for
discipline
: Set of non-available ORs for discipline : Maximum waiting time for low-priority surgeries
{
M: Very large number 4.2 Indexes : Patients : Surgeon : Discipline : Day : Operating room 4.3 Decision variables { { { {
: Start time of surgical case of discipline
: The minimal operability factor of the surgeon that we aim to maximize in order to ensure a fairness workload
: The operability of the surgeon of the discipline during the planning horizon, given by the following formula:
∑ ∑
4.4 Bi-objective Programming model
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Constraints (3) guarantee that a surgical case can be performed at most once. Constraints (4) state an upper limit to surgeries processing time in each OR session. Constraints (5) guarantee that patients of a surgery disci-pline are assigned to an OR session only when the ses-sion is open for that discipline. Constraints (6) guarantee that at most one surgical discipline assigned to an OR on a day. Constraints (7-8) bound the number of weekly OR session assigned to each discipline. Constraints (9) limit the number of parallel OR session assigned to the same surgical discipline. Discipline-to-OR restrictions are considered by constraints (10). Constraints (11) ensure that the dummy patient is assigned to each OR session. Constraints (12)-(13) guarantee the sequencing of the surgeries. Constraints (14) guarantee that one surgeon is assigned at most to an OR session in a day. Availability of surgeons in days is taken into account by constraints (15). Constraints (16) ensure that a surgeon is assigned at most to one OR on a day. Constraints (17) ensure that the workload of each surgeon is greater than the opera-bility factor that we aim to maximize. Constraints (18) compute the start time of surgeries.
4.5 -constraint method:
The main of the -constraint method is to allow the mul-ti-objective problems to be cast as a single objective problem by selecting one of them. The other objective functions will act as the constraints within some speci-fied values.
Moreover, the Pareto Front of a bi-objective problem can be efficiently generated using -constraint (Bérubé et al., 2007); different Pareto optimal solution can be found using different values.
The model (1)-(18) is transformed:
Subject to:
The selected must be in range of . Since the multi-objective problem under study is a bi-objective problem, the -constraint method only need range bound (lower and upper bound) of one objective function.
To generate as many Pareto optimal solutions as possi-ble, the right-hand side of constraint (19) is gradually increased by a small amount and the problem is solved whenever is increased (Dermir et al., 2014).
5 COMPUTATIONAL EXPERIENCE
The database is taking from the archives of a French medium-sized hospital. The OR session is a full day and takes 450min. The hospital has 8 OR, one of them is devoted to the emergencies (EMR). All ORs are equipped with the same materials but some of them are more suitable for some disciplines than others.
The surgeons are grouped according to their discipline and availability.
We have tested the model with different size of the OT; 8 ORs, 12 ORs and 16 ORs and two different patients’ waiting list sizes: 140 and 180 patients. We set
In the following, the notation P.X, R.Y is used to denote a given instance with X patients in the waiting list and Y OR in the OT.
20 surgeons are appointed to treat the patients according to the timetables and disciplines.
Discipline Surgeons Genecology (GYN) {1,2,3} Ambulatory Surgery(DS) {4,5,6,7,8,9,10,11} Urology(URO) {12,13,14} Orthopedic Surgery (ORTH) {15,16,17}
Vascular Surgery (VAS) {18,19,20}
Table1. Surgeons’ discipline
Table 1 gives the surgeons according to their disciplines.
Discipline Genecology {1,4,5,6,7,8} 3 9 1 Ambulatory surgery {8} 4 12 2 Urology {1,2,8} 2 9 1 Orthopedic surgery {2,3,5,6,7,8} 3 11 2 Vascular surgery {1,2,3,4,5,6,8} 4 8 1 Table2. Discipline-to-OR restrictions for instance (P.140, R.8)
Table 2 shows discipline-to-OR restrictions for instance (P.140, R.8).
is the set of OR that can not be used by discipline . For example vascular surgery can only use OR 7. The third and fourth columns give lower and upper number of OR session for each surgery discipline. The last col-umn shows the number of possible parallel session of each discipline.
The model is implemented using CPLEX Optimization Studio 12.6, in an Intel® Core™ i5 and 6 Go Ram. In order to compute the epsilon values for -constraint method, the first step is to compute the (lower and up-per) bounds of the auxiliary objective .
To generate many Pareto optimal solutions for the prob-lem, the right-hand side of constraint (19) is gradually increased by a small amount and the problem is solved whenever increased. (S1) (S2) (S3) (P.140,R.8) 8175 8198 7998 (P.140,R.12) 9388 9236 9253 (P.140,R.16) 10151 10196 10102 (P.180,R.8) 7918 8231 8127 (P.180,R.12) 9454 9436 9512 (P.180,R.16) 10208 10325 10415
Table 3. Numerical results
Numerical results are given in Table 3. We computed three solutions from the Pareto optimal Front.
MONDAY TUESDAY WEDNSDAY THURSDAY FRIDAY R=1 ORTH (15) DS (5) ORTH (17) DS (6) ORTH (16) R=2 DS (11) EMPTY GYN (2) GYN (1) DS (9) R=3 GYN (3) GYN (3) EMPTY DS (11) GYN (2) R=4 ORTH (16) ORTH (17) DS (5) ORTH (17) ORTH (15) R=5 URO (14) URO (13) URO (12) URO (13) DS (7) R=6 DS (4) DS (10) DS (8) EMPTY URO (12) R=7 VAS (20) VAS (18) VAS (20) VAS (19) VAS (19)
R=8 EMR EMR EMR EMR EMR
Table 4. MSS planning for instance (P.140, R.8)
Table 4 shows the results of the MSS and surgeons in each OR session using the proposed approach. The plan-ning respect discipline-to-OR restrictions and resources’ availability. The obtained results show the good balanc-ing of surgeons’ workload.
6 CONCLUSION
In this paper, we proposed a multi-objective program-ming model that aims to balance surgeons’ workload and respect surgeries due dates according to resources’ avail-ability and OT restrictions.
The multi-objective problem is solved using -constraint method and compute several solutions from the Pareto Front. We believe that the approach proposed consists in a good tool to evaluate the quality and efficiency of the OT management. This study could be extended to in-clude downstream hospital resources’ availability to build an efficient planning to the OT.
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